Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups
Fabian Fuchs, Max Nendel
TL;DR
This paper develops a unified framework for the existence of viscosity solutions to abstract Cauchy problems in locally convex vector lattices by introducing $K$-convexity, a generalization of convexity defined via a family $K(t)$ acting on a (potentially) larger lattice $Y$. The main result shows that if a monotone semigroup is $K$-convex with a strongly right-continuous $K$ and its generator $L$ has a nonempty domain $D(L)$, then for any initial data $u_0\in X$, the trajectory $u(t)=S(t)u_0$ is a $D$-$M$-viscosity solution to $u'(t)=Lu(t)$, with sub- and supersolution properties established by semigroup dynamics and limit arguments as $t\downarrow0$. This framework accommodates non-convex inf-sup nonlinearities and enables transferring continuity from the semigroup to the bounding family $K$, while allowing the generator to be analyzed in a larger space $Y$. The paper then demonstrates two infinite-dimensional applications: (i) drift-controlled Lévy dynamics, where the value function solves $u'=Au+c^*(\nabla u)$, and (ii) distributionally robust drift-controlled Ornstein–Uhlenbeck dynamics, yielding a generator involving $\langle x,A^*\nabla u(x)\rangle$, $\sup_{a\in A}\langle b(x,a),\nabla u(x)\rangle$, and $\inf_{\sigma\in\Sigma}\frac{1}{2}\operatorname{tr}(\sigma^*Q\sigma\nabla^2 u(x))$. This advances viscosity-solution theory in infinite-dimensional, uncertain, and non-convex settings with concrete control- and uncertainty-driven models.
Abstract
In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs to satisfy a convexity estimate, so called $K$-convexity, w.r.t. another family of operators, defined on a potentially larger locally convex vector lattice. We then show that, under mild continuity requirements on the bounding family of operators, the semigroup yields viscosity solutions to the abstract Cauchy problem given in terms of its generator in the larger locally convex vector lattice. We apply our results to drift control problems for infinite-dimensional Lévy processes and robust optimal control problems for infinite-dimensional Ornstein-Uhlenbeck processes.